Partial orders of dimension 2
(1972)

Tools

"... Acyclic digraphs, such as the covering digraphs of ordered sets, are usually drawn upward, i.e., with the edges monotonically increasing in the vertical direction. A digraph is upward planar if it admits an upward planar drawing. In this survey paper, we overview the literature on the problem of upw ..."

Acyclic digraphs, such as the covering digraphs of ordered sets, are usually drawn upward, i.e., with the edges monotonically increasing in the vertical direction. A digraph is upward planar if it admits an upward planar drawing. In this survey paper, we overview the literature on the problem of upward planarity testing. We present several characterizations of upward planarity and describe upward planarity testing algorithms for special classes of digraphs, such as embedded digraphs and single-source digraphs. We also sketch the proof of NP-completeness of upward planarity testing.

"... . Temporal Constraint Satisfaction is an information technology useful for representing and answering queries about the times of events and the temporal relations between them. Information is represented as a Constraint Satisfaction Problem (CSP) where variables denote event times and constraints re ..."

. Temporal Constraint Satisfaction is an information technology useful for representing and answering queries about the times of events and the temporal relations between them. Information is represented as a Constraint Satisfaction Problem (CSP) where variables denote event times and constraints represent the possible temporal relations between them. The main tasks are two: (i) deciding consistency, and (ii) answering queries about scenarios that satisfy all constraints. This paper overviews results on several classes of Temporal CSPs: qualitative interval, qualitative point, metric point, and some of their combinations. Research has progressed along three lines: (i) identifying tractable subclasses, (ii) developing exact search algorithms, and (iii) developing polynomial-time approximation algorithms. Most available techniques are based on two principles: (i) enforcing local consistency (e.g. path-consistency), and (ii) enhancing naive backtracking search. Keywords: Temporal Constra...

"... Characterisations of interval graphs, comparability graphs, co-comparability graphs, permutation graphs, and split graphs in terms of linear orderings of the vertex set are presented. As an application, it is proved that interval graphs, cocomparability graphs, AT-free graphs, and split graphs have ..."

Characterisations of interval graphs, comparability graphs, co-comparability graphs, permutation graphs, and split graphs in terms of linear orderings of the vertex set are presented. As an application, it is proved that interval graphs, cocomparability graphs, AT-free graphs, and split graphs have bandwidth bounded by their maximum degree. 1

"... . Starting from a permutation of f0; : : : ; n \Gamma 1g we compute in parallel with a workload of O(n log n) a compact data structure of size O(n log n). This data structure allows to obtain the associated permutation graph and the transitive closure and reduction of the associated order of dimens ..."

. Starting from a permutation of f0; : : : ; n \Gamma 1g we compute in parallel with a workload of O(n log n) a compact data structure of size O(n log n). This data structure allows to obtain the associated permutation graph and the transitive closure and reduction of the associated order of dimension 2 efficiently. The parallel algorithms obtained have a workload of O(m + n log n) where m is the number of edges of the permutation graph. They run in time O(log 2 n) on a CREW PRAM. 1 Introduction Permutation graphs are combinatorial objects that found a lot of attention in recent years. This interest led to many results under a structural point of view as well as algorithmically, see [1, 2, 6, 7, 8]. By definition permutation graphs have a compact encoding of size n, where n is the number of vertices. In sequential computing model, it is possible to pass from the graph to the permutation and vice versa with a workload of O(n 2 ). For parallel computing this has been open up to no...

"... We provide general conditions under which an economic theory has a universal axiomatization: one that leads to testable implications. Roughly speaking, if we obtain a universal axiomatization when we assume that unobservable parameters (such as preferences) are observable, then we can obtain a unive ..."

We provide general conditions under which an economic theory has a universal axiomatization: one that leads to testable implications. Roughly speaking, if we obtain a universal axiomatization when we assume that unobservable parameters (such as preferences) are observable, then we can obtain a universal axiomatization purely on observables. The result &quot;explains&quot; classical revealed preference theory, as applied to individual rational choice. We obtain new applications to Nash equilibrium theory and Pareto optimal choice.

"... . Starting from a permutation of f0; : : : ; n \Gamma 1g we compute in parallel with a workload of O(n log n) a compact data structure of size O(n log n). This data structure allows to obtain the associated permutation graph and the transitive closure and reduction of the associated order of dimens ..."

. Starting from a permutation of f0; : : : ; n \Gamma 1g we compute in parallel with a workload of O(n log n) a compact data structure of size O(n log n). This data structure allows to obtain the associated permutation graph and the transitive closure and reduction of the associated order of dimension 2 efficiently. The parallel algorithms obtained have a workload of O(m + n log n) where m is the number of edges of the permutation graph. They run in time O(log 2 n) on a CREW PRAM. 1. Introduction Permutation graph are combinatorial objects that found a lot of attention in recent years. This interest led to many results under a structural point of view as well as algorithmically, see [BFR71, Col81, Gol85, PLE71, SV83]. By definition permutation graphs have a compact encoding of size n, where n is the number of vertices. In sequential computing model, it is possible to pass from the graph to the permutation and vice versa with a workload of O(n 2 ). For parallel computing this ha...

"... We show that a transitively reduced digraph has a confluent upward drawing if and only if its reachability relation has order dimension at most two. In this case, we construct a confluent upward drawing with O(n2) features, in an O(n)×O(n) grid in O(n2) time. For the digraphs representing series-par ..."

We show that a transitively reduced digraph has a confluent upward drawing if and only if its reachability relation has order dimension at most two. In this case, we construct a confluent upward drawing with O(n2) features, in an O(n)×O(n) grid in O(n2) time. For the digraphs representing series-parallel partial orders we show how to construct a drawing with O(n) features in an O(n)×O(n) grid in O(n) time from a series-parallel decomposition of the partial order. Our drawings are optimal in the number of confluent junctions they use. Submitted:

"... We consider the problem of routing messages on networks modeled by Series Parallel Graphs (SPGs), and we introduce a new technique, called Distance Routing (DR). We first present an algorithm that computes shortest path DR on directed SPGs, and we show how to apply to these networks the shortest pat ..."

We consider the problem of routing messages on networks modeled by Series Parallel Graphs (SPGs), and we introduce a new technique, called Distance Routing (DR). We first present an algorithm that computes shortest path DR on directed SPGs, and we show how to apply to these networks the shortest path 1-Interval Routing, one of the most common compact routing techniques. We also compute and compare the complexities of these two techniques showing the strong advantage of DR especially in terms of time complexity. We then show how Distance Routing can be used to route on bidirectional SPGs, where no general shortest path 1-Interval Routing Scheme can be applied. Index Terms: Routing, Compact Routing Algorithms, Distance Routing, Networks, Series Parallel Networks. 1 Introduction In a non anonymous network (i.e., in a network where to each node it is associated a different identity) routing can be easily accomplished if each node has available a routing table. This table has n \Gamma 1 e...